Sync FASTMAIL contacts & calendars with Outlook/Office365

As a fan of Fastmail, I often use their mail, calendar, and contacts through the webmail interface. However, I also want to use Fastmail's contacts and calendars in Outlook. How can I achieve this? With the ECO plugin for Outlook, you can sync Fastmail's contacts and calendars via the CardDAV and CalDAV protocols. Let's see how to do it.